This paper addresses the problems of reachable set estimation and controller design for two-dimensional (2-D) singular Roesser systems with time-varying delays. We first decompose the system into dynamic- and algebraic-subsystems and formulate a bounding ellipsoid that contains the reachable set of the system. Based on a 2-D Lyapunov functional scheme, and by utilizing zero-type matrix equations, we then derive tractable algebraic conditions to ensure that the system is admissible and admits an ellipsoid that bounds the set of all system states subject to bounded exogenous disturbances. The design problem of a state-feedback controller that makes the closed-loop reachable set is confined within a prescribed ellipsoid is also addressed. The efficacy of the proposed method is demonstrated by numerical examples with simulations.
KaczorekK.Two-dimensional linear systems. Berlin: Springer-Verlag, 1985.
2.
BorsDWalczakS.Application of 2D systems to investigation of a process of gas filtration. Multidimens Syst Signal Process2012; 23(1–2): 119–130.
3.
RogersEGalkowskiKPaszkeW, et al. Multidimensional control systems: case studies in design and evaluation. Multidimens Syst Signal Process2015; 26(4): 895–939.
4.
RoesserR.A discrete state-space model for linear image processing. IEEE Trans Automat Contr1975; 20(1): 1–10.
5.
AhnCKShiPBasinMV.Two-dimensional dissipative control and filtering for Roesser model. IEEE Trans Automat Contr2015; 60(7): 1745–1759.
6.
BachelierOYeganefarNMehdiD, et al. On stabilization of 2D Roesser models. IEEE Trans Automat Contr2017; 62(5): 2505–2511.
7.
ZhangGWangW.Finite-region stability and finite-region boundedness for 2D Roesser models. Math Methods Appl Sci2016; 39(18): 5757–5769.
8.
FornasiniEMarchesiniG.State-space realization theory of two-dimensional filters. IEEE Trans Automat Contr1976; 21(4): 484–492.
9.
FornasiniEMarchesiniG.Doubly-indexed dynamical systems: state-space models and structural properties. Math Syst Theory1978; 12(1): 59–72.
10.
FornasiniEMarchesiniG.On the internal stability of two dimensional filters. IEEE Trans Automat Contr1979; 24(1): 129–130.
11.
ZhangGTrentelmanHLWangW, et al. Input-output finite-region stability and stabilization for discrete 2-D Fornasini-Marchesini models. Syst Control Lett2017; 99: 9–16.
12.
LiDLiangJWangF.Dissipative networked filtering for two-dimensional systems with randomly occurring uncertainties and redundant channels. Neurocomputing2019; 369: 1–10.
13.
TaoYYWuZG.Asynchronous control of two-dimensional Markov jump Roesser systems: an event triggering strategy. IEEE Trans Netw Sci Eng2022; 9(4): 2278–2289.
14.
LiuJYangYSongY.Asynchronous event-triggered guaranteed cost control of uncertain 2-D Markov jump Roesser systems with actuator saturation. J Syst Control Eng2024; 238(8): 1397–1409.
15.
MalikSHTufailMRehanM, et al. Overflow oscillations-free realization of discrete-time 2D Roesser models under quantization and overflow constraints. Asian J Control2022; 24(3): 1416–1425.
16.
FuJDuanZXiangZ.On mixed fault detection observer design for positive 2D Roesser systems: necessary and sufficient conditions. J Franklin Inst2022; 359(1): 160–177.
17.
HienLVVuLHTrinhH.Stability of two-dimensional descriptor systems with generalized directional delays. Syst Control Lett2018; 112: 42–50.
18.
PengDNieH.Abel lemma-based finite-sum inequality approach to stabilization for 2-D time-varying delay systems. Asian J Control2021; 23(3): 1394–1406.
19.
YangRLiLShiP.Dissipativity-based two-dimensional control and filtering for a class of switched systems. IEEE Trans Syst Man Cybern Syst2021; 51(5): 2737–2750.
20.
YangRDingSZhengWX.Co-design of event-triggered mechanism and dissipativity-based output feedback controller for two-dimensional systems. Automatica2021; 130: 109694.
21.
LiLYangRFengZ, et al. Dissipative filtering for two-dimensional LPV systems: a hidden Markov model approach. ISA Trans2022; 130: 409–419.
22.
HeKSunLYangR.Robust stabilization and adaptive control of nonlinear singular systems via static output feedback. Proc IMechE, Part I: J Systems and Control Engineering2024; 238(1): 87–96.
23.
SuYYangDRenJ.Event-triggered control for uncertain discrete-time singular systems via sliding mode approach. Proc IMechE, Part I: J Systems and Control Engineering2024; 238(8): 1459–1472.
24.
XuHZouY.Control for 2-D singular delayed systems. Int J Syst Sci2011; 42(4): 609–619.
25.
ZouYXuHWangW.Stability for two-dimensional singular discrete systems described by general model. Multidimens Syst Signal Process2008; 19(2): 219–229.
26.
Van HienLVuLHPhatVN. Improved delay-dependent exponential stability of singular systems with mixed interval time-varying delays. IET Control Theory Appl2015; 9(9): 1364–8644.
27.
LamJZhangBChenY, et al. Reachable set estimation for discrete-time linear systems with time delays. Int J Robust Nonlinear Control2015; 25(2): 269–281.
28.
FengZLamJ.An improved result on reachable set estimation and synthesis of time-delay systems. Appl Math Comput2014; 249: 89–97.
29.
WangWZhongSLiuF, et al. Reachable set estimation for linear systems with time-varying delay and polytopic uncertainties. J Franklin Inst2019; 356(13): 7322–7346.
30.
ShenJLamJ.Improved reachable set estimation for positive systems: a polyhedral approach. Automatica2021; 124: 109167.
31.
ChenYLamJZhangB.Estimation and synthesis of reachable set for switched linear systems. Automatica2016; 63: 122–132.
32.
BalandinDVBiryukovRSKoganMM.Ellipsoidal reachable sets of linear time-varying continuous and discrete systems in control and estimation problems. Automatica2020; 116: 108926.
33.
FengZLamJ.On reachable set estimation of singular systems. Automatica2015; 52: 146–153.
34.
FengZLiWLamJ.Dissipativity analysis for discrete singular systems with time-varying delay. ISA Trans2016; 64: 86–91.
35.
LiJFengZZhangC.Reachable set estimation for discrete-time singular systems. Asian J Control2017; 19(5): 1862–1869.
36.
LiJZhaoYFengZ, et al. Reachable set estimation and dissipativity for discrete-time T-S fuzzy singular systems with time-varying delays. Nonlinear Anal Hybrid Syst2019; 31: 166–179.
37.
ZhangLFengZJiangZ, et al. Improved results on reachable set estimation of singular systems. Appl Math Comput2020; 385: 125419.
38.
FengZZhangXLamJ, et al. Estimation of reachable set for switched singular systems with time-varying delay and state jump. Appl Math Comput2023; 456: 128132.
39.
FengZZhangXLiuJJR, et al. A new method of reachable sets estimation for the nonlinear switched singular system with impulsive performance and time-delay. Appl Math Comput2024; 481: 128929.
40.
FengZZhangXLiuJJR, et al. Estimation of real-time reachable set on nonlinear time-delay switched singular systems with state jump. Int J Robust Nonlinear Control2024; 34(16): 11199–11217.
41.
TrinhHHienLV.On reachable set estimation of two-dimensional systems described by the Roesser model with time-varying delays. Int J Robust Nonlinear Control2018; 28(1): 227–246.
42.
MaHTianDLiM, et al. Reachable set estimation for 2-D switched nonlinear positive systems with impulsive effects and bounded disturbances described by the Roesser model. Math Model Control2024; 4(2): 152–162.
43.
Van HienLAnNTTrinhH. New results on state bounding for discrete-time systems with interval time-varying delay and bounded disturbance inputs. IET Control Theory Appl2014; 8(14): 1405–1414.
